Finite Group Databasebeta

← #13147762

Subgroups — #13147762

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6229C211yesmaximal, characteristic, complemented
2#64081⋊C3C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721112C212yesmaximal
5#11721109C212yesmaximal
6#11721107C211yesmaximal, characteristic
7#11721106C211yesmaximal, characteristic
8#11721098C212yesmaximal, complemented
9#11721093C211yesmaximal, characteristic, complemented
10#11721091C212yesmaximal, complemented
11#11721090C211yesmaximal, characteristic, complemented
12#10593883—31nomaximal
13#10593880—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C42×H(3,𝔽3)C2211yescharacteristic
18C4⋊#1084C2211yescharacteristic
19#88540C2212yes—
20C4×#1084C2212yes—
21#3752⋊C3C2211yescharacteristic
22#3752⋊C3C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24#3750⋊C3C2211yescharacteristic
25#3750⋊C3C2211yescharacteristic
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2212yes—
29#88450C2212yes—
30#88450C2212yes—
31#88449C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37#64081D311yescharacteristic
38C4×#634D311yescharacteristic, complemented
39C3×#872—31no—
40C3×C4×Dic6—31no—
41C3×C4×Dic6—31no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—61no—
44#63873—31no—
45#63872—31no—
46#63871—31no—
47#63871—31no—
48#63870—32no—
49#63869—31no—
50#63868—32no—
51#63868—32no—
52#63867—32no—
53#63867—32no—
54#63867—32no—
55#63866—31no—
56#63862—32no—
57#63859—32no—
58#63857—31no—
59#63856—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2312yes—
66#6229—22no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2312yes—
72C4×(C3⋊D3)⋊C3—22no—
73#634⋊C3—22no—
74#5583—91no—
75#5583—91no—
76#5518—91no—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×#634C2×D311yescharacteristic
79C2×C3×C4×D3—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C32×C42—21no—
85C32×C42C2×D311yescharacteristic
86C32×C42C2×D311yescharacteristic
87#3752C2×D311yescharacteristic
88#3752C2×D311yescharacteristic
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90#3750C2×D312yes—
91#3750C2×D311yescharacteristic
92#3750C2×D311yescharacteristic
93C4×C3⋊Dic3C2×D312yes—
94C4×C3⋊Dic3C2×D311yescharacteristic
95C3×(C2×D3)⋊C4—31no—
96C3×(C2×D3)⋊C4—31no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—61no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—31no—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—31no—
106C3×Dic3⋊C4—62no—
107C3×C4×Dic3—32no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—32no—
111C3×C4×Dic3—62no—
112#3724—32no—
113#3724—32no—
114#3724—32no—
115#3724—32no—
116Dic3⋊Dic3—32no—
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124#3722—32no—
125#3722—32no—
126(C2×D3)⋊Dic3—32no—
127(C2×D3)⋊Dic3—32no—
128C22×H(3,𝔽3)C2411yescharacteristic
129C2×(C3⋊D3)⋊C3—21no—
130C2×(C3⋊D3)⋊C3—21no—
131C2×(C3⋊D3)⋊C3—21no—
132C4×H(3,𝔽3)—21no—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)#4112yes—
135#1084—22no—
136#1084—22no—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—22no—
142C3×C4×Q8—91no—
143C3×C4×Q8—91no—
144C3×#117—91no—
145#946—91no—
146#946—91no—
147#944—91no—
148#944—91no—
149#924—91no—
150#924—91no—
151#898—92no—
152#898—92no—
153#897—91no—
154#897—91no—
155#890—91no—
156#890—91no—
157#889—92no—
158#889—92no—
159#889—92no—
160#889—92no—
161#879—92no—
162#879—92no—
163#876—92no—
164#876—92no—
165#872—31no—
166#872—91no—
167#870—92no—
168#870—92no—
169#870—91no—
170#869—91no—
171C4×Dic6—31no—
172C4×Dic6—31no—
173C4×Dic6—31no—
174C4×Dic6—61no—
175C4×Dic6—91no—
176C22×C3⋊D3C22×D311yescharacteristic
177C22×C3×D3—31no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C4×C3⋊D3—22no—
193#634—22no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—32no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—62no—
204C2×C3×Dic3—62no—
205C3×C4×D3—62no—
206C3×Dic6—64no—
207C3×Dic6—62no—
208C3×Dic6—62no—
209C3×Dic6—62no—
210#541—271noSylow
211C2×H(3,𝔽3)C2×#4111yescharacteristic
212C2×H(3,𝔽3)2−1+411yescharacteristic
213C2×H(3,𝔽3)2−1+411yescharacteristic
214(C3⋊D3)⋊C3—41no—
215C2×C3×Q8—91no—
216C2×C3×Q8—91no—
217C22×C3×C4—91no—
218C2×C4×D3—31no—
219C2×C4×D3—91no—
220C2×Dic6—31no—
221C2×Dic6—31no—
222C2×Dic6—31no—
223C2×Dic6—61no—
224C2×Dic6—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—92no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4—91no—
231C3×C22⋊C4—91no—
232C3×C22⋊C4—91no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—61no—
236C3×C42—91no—
237C3×C42—92no—
238C3×C42—92no—
239C3×C42D3211yescharacteristic
240#217—92no—
241#217—92no—
242(C2×D3)⋊C4—31no—
243(C2×D3)⋊C4—31no—
244(C2×D3)⋊C4—91no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4—92no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—92no—
249(C2×D3)⋊C4—92no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—31no—
253C4⋊Dic3—61no—
254C4⋊Dic3—91no—
255C4⋊Dic3—92no—
256C4⋊Dic3—92no—
257C4⋊Dic3—92no—
258C4⋊Dic3—92no—
259C4⋊Dic3—92no—
260C4⋊Dic3—92no—
261Dic3⋊C4—32no—
262Dic3⋊C4—32no—
263Dic3⋊C4—32no—
264Dic3⋊C4—31no—
265Dic3⋊C4—31no—
266Dic3⋊C4—62no—
267Dic3⋊C4—92no—
268Dic3⋊C4—91no—
269Dic3⋊C4—91no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282C4×Dic3—32no—
283C4×Dic3—32no—
284C4×Dic3—32no—
285C4×Dic3—31no—
286C4×Dic3—62no—
287C4×Dic3—92no—
288C4×Dic3—91no—
289C22×C32—21no—
290C22×C32C23×D311yescharacteristic
291C22×C32C23×D311yescharacteristic
292C2×C3⋊D3—21no—
293C2×C3⋊D3—21no—
294C2×C3⋊D3—21no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C32×C4—22no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—22no—
304C32×C4—41no—
305C32×C4#23512yes—
306C32×C4#23512yes—
307C3⋊Dic3—22no—
308C3⋊Dic3—21no—
309C3⋊Dic3—21no—
310C3⋊Dic3—21no—
311C3⋊Dic3—22no—
312C3×Dic3—64no—
313C3×Dic3—62no—
314C3×Dic3—61no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—62no—
319C3×Dic3—62no—
320C3×Dic3—61no—
321C3×Dic3—62no—
322C3×Dic3—122no—
323C4⋊Q8—271no—
324#126—272no—
325#126—272no—
326#125—272no—
327#125—272no—
328#125—271no—
329C22⋊Q8—271no—
330C22⋊Q8—271no—
331C4×Q8—91no—
332C4×Q8—91no—
333#117—91no—
334H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
335C23×C3—91no—
336C22×D3—31no—
337C22×D3—91no—
338C3×Q8—182no—
339C3×Q8—182no—
340C2×C3×C4—31no—
341C2×C3×C4—31no—
342C2×C3×C4—31no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—61no—
347C2×C3×C4—61no—
348C2×C3×C4—61no—
349C2×C3×C4—91no—
350C2×C3×C4—92no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—92no—
356C2×C3×C4—182no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360C2×Dic3—32no—
361C2×Dic3—32no—
362C2×Dic3—32no—
363C2×Dic3—32no—
364C2×Dic3—31no—
365C2×Dic3—32no—
366C2×Dic3—31no—
367C2×Dic3—32no—
368C2×Dic3—31no—
369C2×Dic3—62no—
370C2×Dic3—62no—
371C2×Dic3—91no—
372C2×Dic3—92no—
373C2×Dic3—91no—
374C2×Dic3—92no—
375C2×Dic3—91no—
376C4×D3—62no—
377C4×D3—182no—
378Dic6—62no—
379Dic6—62no—
380Dic6—64no—
381Dic6—62no—
382Dic6—182no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32C2×#23511yescharacteristic
387C2×C32C2×#23511yescharacteristic
388C2×C32#100711yescharacteristic
389C2×C32#101011yescharacteristic
390C2×C32#101011yescharacteristic
391C2×C32#101011yescharacteristic
392C3⋊D3—41no—
393C3×D3—121no—
394C2×Q8—91no—
395C2×Q8—91no—
396C22×C4—91no—
397C4⋊C4—91no—
398C4⋊C4—91no—
399C4⋊C4—91no—
400C4⋊C4—91no—
401C4⋊C4—92no—
402C4⋊C4—92no—
403C4⋊C4—272no—
404C4⋊C4—272no—
405C4⋊C4—272no—
406C4⋊C4—272no—
407C4⋊C4—272no—
408C4⋊C4—272no—
409C22⋊C4—91no—
410C22⋊C4—91no—
411C22⋊C4—272no—
412C22⋊C4—272no—
413C42—91no—
414C42—92no—
415C42—92no—
416C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—61no—
420C22×C3—181no—
421C22×C3—181no—
422C22×C3—181no—
423C22×C3C22×D3211yescharacteristic
424C2×D3—61no—
425C2×D3—61no—
426C2×D3—61no—
427C2×D3—181no—
428C2×D3—181no—
429C2×D3—181no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—32no—
433C3×C4—32no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—62no—
439C3×C4—62no—
440C3×C4—121no—
441C3×C4—181no—
442C3×C4—182no—
443C3×C4—181no—
444C3×C4—181no—
445C3×C4—182no—
446C3×C4—182no—
447C3×C4—182no—
448C3×C4#379812yes—
449Dic3—62no—
450Dic3—62no—
451Dic3—62no—
452Dic3—62no—
453Dic3—64no—
454Dic3—61no—
455Dic3—62no—
456Dic3—61no—
457Dic3—62no—
458Dic3—61no—
459Dic3—122no—
460Dic3—182no—
461Dic3—181no—
462Dic3—181no—
463Dic3—182no—
464Dic3—181no—
465C32—21no—
466C32#551811yescharacteristic, complemented
467C32#558311yescharacteristic, complemented
468C23—91no—
469Q8—182no—
470Q8—182no—
471C2×C4—91no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—92no—
476C2×C4—92no—
477C2×C4—91no—
478C2×C4—182no—
479C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
480C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
481C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—61no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—361no—
492C2×C3#6429711yescharacteristic
493C2×C3#6429811yescharacteristic
494C2×C3C2×#379811yescharacteristic
495D3—121no—
496D3—361no—
497C22—181no—
498C22—181no—
499C22—181no—
500C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C4—21no—
502C4—21no—
503C4—182no—
504C4—181no—
505C4—181no—
506C4—182no—
507C4—181no—
508C4—182no—
509C4—182no—
510C4#8865012yes—
511C3—31no—
512C3—31no—
513C3—61no—
514C3#1059388011yescharacteristic
515C2—361no—
516C2#1172248211yescharacteristic
517C2#1172248311yescharacteristic
518C2C2×#8865011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C4×#6229C2semidirect
3. C4×#634⋊C3C2semidirect
8. #11721098C2semidirect
9. #11721093C2semidirect
10. #11721091C2semidirect
11. #11721090C2semidirect
38. C4×#634D3semidirect
334. H(3,𝔽3)#541semidirect
466. C32#5518semidirect
467. C32#5583semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C4×#62292
#64081⋊C32
C4×#634⋊C32
#117211122
#117211092
#117211072
#117211062
#117210982
#117210932
#117210912
#117210902
#105938833
#105938803